The Lp Minkowski problem for polytopes for p<0

Abstract

Abstract. Necessary and sufficient conditions are given for the existence of solutions to the discrete L p Minkowski problem for the critical case where 0 < p < 1.

The L dual Minkowski problem for p > 1 and q > 0

“…We also note that if p ≤ 0 and µ is discrete such that any n elements of suppµ are independent vectors, then µ = S p (K, ·) = n · C p,n (P, ·) for some polytope P ∈ K n (o) according to Zhu [30,31]. In this paper, we first solve the discrete L p dual Minkowski problem if p > 1 and q > 0.…”

On the polar Orlicz-Minkowski problems and the p-capacitary Orlicz-Petty bodies

“…However, as showed in, e.g. [59,60], the solutions to the L q Minkowski problems for discrete measures could be well-existed for q < 0. Finally, it seems intractable to find a direct relation between µ ∈ Ω and the solutions to the polar Minkowski problems, while such a relation usually can be established as long as the solutions exist for the related Minkowski problems.…”

“…In this case, especially if K ∈ K 0 , one can link µ to a convex body. However, it is not clear whether, in general, there exists a convex body K ∈ K 0 such that dµ = c · h 1−q K dS(K, ·) for q < 1; see special cases in [16,58,59,60]. In other words, µ ∈ Ω, although closely related to convex bodies, is in fact more general than the measures generated from convex bodies.…”

On the general dual Orlicz-Minkowski problem

“…It has been proved in [34, Propositions 3.1 and 3.3] that the solutions to the polar Orlicz-Minkowski problem for discrete measures must be polytopes, the convex hulls of finite points in R n . It is well-known that all convex bodies can be approximated by polytopes, and hence to study the Minkowski type problems for discrete measures is very important and receives extensive attention, see e.g., [2,3,11,15,21,23,26,29,30,53,65,66,67].…”